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**Chapter 19: Flexible Machine Elements**

Scientists study the world as it is, engineers create the world that never has been. Theodore von Karman A rolling chain on a sprocket. Source: Shutterstock.

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**Comparison of Power Transmission Devices**

Table 19.1: Comparison of selected power transmission devices.

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Belt Dimensions Figure 19.1: Dimensions, angles of contact, and center distance of open flat belt.

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Weighted Idler Figure 19.2: Weighted idler used to maintain desired belt tension.

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Synchronous Belt Figure 19.3: Synchronous, or timing, belt.

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V-Belts Figure 19.4: Design and construction of V-belts. (a) Standard V-belts cross sections with dimensions; (b) Typical single-belt, showing reinforcing cords and wear resistant exterior; (c) Double V-belt, used for higher power transmission than single belts. Up to five belts can be combined in this fashion.

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V-Belt in Groove Figure 19.5: V-belt in sheave groove.

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Power by Belt Class Figure 19.6: Guide to selection of belt cross section as a function of power transmitted and shaft speed.

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Overload Factors Table 19.2: Typical overload service factors, f1. Source: Courtesy of Gates Corp.

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**Minimum Datum Diameters**

Table 19.3: Recommended minimum datum diameters, in inches, of sheaves for general purpose 60-cycle electric motors. Source: Courtesy of Gates Corp.

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**Design Procedure 19.1: V-Belt Drives**

It will be assumed that a belt drive will be designed for power transmission where the shaft speeds (and hence speed ratio) and desired center distance are known. The power available can be obtained from the rating of the motor, or else it can be obtained from design requirements. Based on these quantities, this design procedure provides a methodology for selecting a cross-section of a belt, choosing sheaves and number of belts required. Estimate the overload service factor from Table 19.2 and use it to obtain the required belt power rating using Eq. (19.11). Select a cross section of the belt from the required belt power rating and the shaft speed using Fig Obtain the minimum allowable sheave datum diameter from Table 19.3. Locate the sheave diameter combinations in Table 19.4 that are suitable for a desired speed ratio. Disregard from consideration any candidates that are smaller than the minimum values obtained in Step 3. From the remaining candidates, select a sheave size that is consistent with space requirements.

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**Design Procedure 19.1 (concluded)**

From Table 19.4, locate the center distance that most closely matches design constraints, and obtain the power correction factor, f2. Note that the belt length can be calculated from Eq. (19.5) or read directly from Table 19.4. From Table 19.5, locate the proper belt cross section and center distance, and obtain the basic power rating per belt, h1. Note that for very high speeds or small sheaves, an additional power may be required. This is usually a small amount and is neglected in this design procedure. The rated power per belt is given by The number of belts required can be obtained from the required power from Step 1:

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Center Distance Table 19.4: Center distance and power correction factor, f2, for standard sheaves. Source: Courtesy of Gates Corp.

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**Rated Horsepower per Belt**

Table 19.5: Rated power in horsepower per belt for selected 3V and 5V cross sections. Source: Courtesy of Gates Corp.

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Wire Rope Figure 19.7: Cross sections of selected wire rope. (a) 6 x 19 fiber core; (b) 1 x 19; (c) 6 x 36 wire core; (d) 18 x 7 fiber core. Figure 19.8: Two lays of wire rope. (a) Lang; (b) regular.

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Wire Rope Data Table 19.6: Wire rope data. Source: From Shigley and Mitchell [1983].

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**Sheave to Rope Diameter Effects**

Figure 19.9: Percent strength loss in wire rope for different D/d ratios. Figure 19.10: Service life for different D/d ratios.

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Safety Factors Table 19.7: Minimum safety factors for a variety of wire rope applications. Note that the use of these safety factors does not preclude a fatigue failure. Source: From Shigley and Mitchell [1983].

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**Maximum Bearing Pressure**

Table 19.8: Maximum allowable bearing pressures for various sheave materials and types of rope Source: From Shigley and Mitchell [1983]

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**Rolling Chain Figure 19.11: Various parts of rolling chain.**

Figure 19.12: Typical rolling chain. (a) One-strand rolling chain; (b) three-strand chain.

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**Standard Rolling Chain Sizes**

Table 19.9: Standard sizes and strengths of rolling chains.

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Chordal Rise Figure 19.13: Chordal rise in rolling chains. Note that the chain link travels upwards as well as horizontally when moving from position A to position B.

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**Service and Multiple Strand Factors**

Table 19.10: Service factor, a1, for rolling chains. Table 19.11: Multiple-strand factor for rolling chains.

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**Power Rating for Rolling Chains**

Table 19.12: Power rating of selected standard roller chains, in horsepower.

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Chain Power Ratings Figure 19.14: Design guideline for standard roller chains.

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Silent Chain Figure 19.16: The use of guide links in silent chains. (a) One guide link in center of chain; (b) two center guide links; (c) Two side guide links. Source: Courtesy of Ramsey Products Corp. Figure 19.15: A silent chain drive. (a) Silent chain with sprockets; (b) detail of silent chain links. Source: Courtesy of Ramsey Products Corp.

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**Design Procedure 19.2: Design of Chain Drives**

For the purposes of this Design Procedure, the power transmitted (or chain force and speed), power source, speed ratio, and loading environment need to be known, or at least be somewhat constrained. Obtain the service factor from Table Calculate the chain's required power rating from Eq. (19.31), taking a2=1.0. Select a chain size from Fig using the required power rating and the small sprocket speed. Note that using the fewest number of chain strands while satisfying power requirements usually results in the most economical design. Obtain the strand factor, a2, from Table The required power rating, given by Eq. (19.31), needs to be recalculated if a multiple strand chain is to be used. Referring to Table 19.12, identify the column of the table that corresponds to the small sprocket's speed. Reading down from the top, find the number of teeth in the smaller sprocket that produces the required modified power rating, h’pr. This is the minimum number of teeth that are required for the application. Larger sprockets can be used if desired.

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**Design Procedure 19.2 (concluded)**

If a multiple strand chain is being considered, record the modified power rating from Table 19.12, and multiply by the strand factor to obtain the chain's power rating. Note the required lubrication method in Table Variation from the lubrication approach may compromise chain longevity. The number of teeth on the larger sprocket can be calculated from the desired velocity ratio by using Eq.~(19.22). If the center distance has not been prescribed, it can be estimated by recognizing that cd/pt should be between 30 and 50, although larger lengths can be allowed if chain guides are incorporated into the design. If the center distance exceeds space limitations, increase the number of strands or select the next largest pitch chain and return to Step 4. The number of links in the chain can be calculated from Eq. (19.26), rounded up to the next highest even integer.

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**Case Study: Drag Line Gantry**

Figure 19.17: Typical dragline.

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